Can I export a trained machine learning model so that others case use it?
If the 'algorithm' you're talking about is a neural network, then you can distribute the learned parameters/weights to anyone who wants to use it. This is how neural nets are normally 'exported': without all of the training data used to create them. Actually, this is done with many kinds of models (parameterized ones). 

In order to 'decode' the model, users would only have to know its structure. In the case of a neural network, they'd need to know the size of each layer, what activation functions were used, etc. 

This is not possible with every type of ML model, however. Specifically, non-parametric models and 'lazy' models make use of training data at inference time. They wouldn't be useful without their training data. Classifying an input by finding its k nearest neighbors, for example, would require having the training data.